![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > apcon4bid | GIF version |
Description: Contrapositive law deduction for apartness. (Contributed by Jim Kingdon, 31-Jul-2023.) |
Ref | Expression |
---|---|
apcon4bid.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
apcon4bid.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
apcon4bid.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
apcon4bid.d | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
apcon4bid.1 | ⊢ (𝜑 → (𝐴 # 𝐵 ↔ 𝐶 # 𝐷)) |
Ref | Expression |
---|---|
apcon4bid | ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | apcon4bid.1 | . . 3 ⊢ (𝜑 → (𝐴 # 𝐵 ↔ 𝐶 # 𝐷)) | |
2 | 1 | notbid 639 | . 2 ⊢ (𝜑 → (¬ 𝐴 # 𝐵 ↔ ¬ 𝐶 # 𝐷)) |
3 | apcon4bid.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
4 | apcon4bid.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
5 | apti 8296 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵)) | |
6 | 3, 4, 5 | syl2anc 406 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵)) |
7 | apcon4bid.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
8 | apcon4bid.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
9 | apti 8296 | . . 3 ⊢ ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (𝐶 = 𝐷 ↔ ¬ 𝐶 # 𝐷)) | |
10 | 7, 8, 9 | syl2anc 406 | . 2 ⊢ (𝜑 → (𝐶 = 𝐷 ↔ ¬ 𝐶 # 𝐷)) |
11 | 2, 6, 10 | 3bitr4d 219 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1312 ∈ wcel 1461 class class class wbr 3893 ℂcc 7539 # cap 8255 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-mulrcl 7638 ax-addcom 7639 ax-mulcom 7640 ax-addass 7641 ax-mulass 7642 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-1rid 7646 ax-0id 7647 ax-rnegex 7648 ax-precex 7649 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-lttrn 7653 ax-pre-apti 7654 ax-pre-ltadd 7655 ax-pre-mulgt0 7656 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-iota 5044 df-fun 5081 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-pnf 7720 df-mnf 7721 df-ltxr 7723 df-sub 7852 df-neg 7853 df-reap 8249 df-ap 8256 |
This theorem is referenced by: mul0eqap 8338 abs00 10722 mul0inf 10898 |
Copyright terms: Public domain | W3C validator |